Fundamentals of Pattern Recognition Book

	Preface to the Second Edition
	Preface to the First Edition
1	Introduction	1
1.1	Organization of the Book	1
1.2	Mathematical Methods and Their Ordering in Fundamental Research, Particularly in Pattern Recognition	2
2	The Fundamental Problem of Pattern Recognition	5
2.1	Classification and Recognition: General Statement of the Fundamental Problem of Mathematics and Some Examples	5
2.2	Isomorphism and Decidability: The "Word Problem"	7
2.3	Consequences upon Pattern Recognition's Theory, Tools, and Methodology: The Fundamental Problem of Pattern Recognition. Approximation of the Given Problem by a Decidable and Automatically Computable One	8
3	Images, Fiber Images, and Shapes: The Topological Framework	13
3.1	The Discretization Problem	13
3.1.1	Different Topologies on a Finite Set, Their Computer Enumeration, and Their Machine Representation	15
3.1.2	Finite Topological Spaces: The General Topological Discussion	19
3.2	Definition, Classification, and Recognition of Images in Discrete or Digitized Spaces	26
3.2.1	Finitely Presented Abelian Groups: Discrete or Digitized Spaces	29
3.2.2	Paths, Metrics, Neighborhoods	36
3.2.3	Dimension in Discrete Spaces	39
3.2.4	Connectivity: Decidability; Connected Components, Definition of a Discrete Topological Similarity; Classification and Recognition	45
3.3	The Mathematical Theory of Shape	54
3.3.1	The Notion of Shape	55
3.3.2	The Notion of Shape for Very Irregular Spaces	61
3.3.3	Shape Invariants, Operations, and Properties	67
3.3.4	Interest of the Theory of Shape for Automatic Pattern Recognition: Impact, Limitations, Open Problems	70
3.4	The General Topological Formalization of Images and Patterns or Shapes: The Fiberwise Topological Framework	75
3.4.1	Questions About the Completeness of Images' Representation; A Shortcoming Common to all Previous Formalisms: Filling the Gap by Fiberwise Topology	75
3.4.2	Preliminary Basic Notions, Glossary, and Some Results in Fiberwise Topology: The "Homotopy Connection." Fiberings, Bundles, Slices, Fiberwise Topological Spaces	79
3.4.3	Pattern Recognition in Fiberwise Topology: Fiber Images, Categories of Fiber Images, Shapes or Topological Patterns of Fiber Images	88
3.4.4	Interest of Fiberwise Topology for Automatic Pattern Recognition: Impact, Limitations, Open Problems, Concluding Remarks	110
4	Objects or Images, Structures or Patterns, Classification and Recognition: The Structural or Syntactic Framework	117
4.1	Academic Versus Real Images	117
4.2	Abstract Objects: Syntactic Generation of Images	122
4.2.1	Hypotheses on Images and on the Transformations They Undergo	124
4.2.2	Grammars of Images	130
4.2.3	Definition of Structures or Patterns	143
4.3	Images of the Real World: Deformed Images and Their Recognition Problem	148
4.3.1	Deformation (Image) Grammars	149
4.3.2	Setting of the Classification and Recognition Problems; Metrics, Metric Deformation Grammars	154
4.4	The Decision Problem for Automata	164
4.4.1	Correspondence Between the Decision Problem for Automata and Pattern Recognition	165
5	General Formalization of the Pattern Recognition Problem: The Categorical Framework	169
5.1	The Necessity of Enlarged Mathematical Hypotheses: The Interest of the Categorical Approach	169
5.1.1	Unification of the Topological and Algebraic Approaches	171
5.1.2	Revisiting Real Images or Objects	174
5.2	Preliminary Notions About Categories, Functors, Invariant Functors, Natural Transformations, and Equivalence	175
5.3	Pattern Recognition in Categories	189
5.3.1	Image Categories	189
5.3.2	Recognition Categories; Patterns	192
5.3.3	Probes and Recognition Functions	194
5.4	Comparison of Image Categories	198
5.4.1	Retracts or Projections of Images	198
5.4.2	"Order" of Complexity of Image Categories	200
5.4.3	Skeletons	203
5.4.4	Interest of the Theory of Categories for Automatic Pattern Recognition: Impact Limitations, Open Problems	211
6	Conclusion	223
6.1	Synthesis of This Work	224
6.2	Open Problems in Automatic Pattern Recognition	226
	Appendices	231
	References	237
	Index	243